There is a simple way we can calculate the Coriolis force on a body moving zonally (i.e., in the east-west direction) if we already know the form of the centrifugal force. For simplicity, we first suppose that Earth is a flat disk, with the axis of rotation perpendicular to the disk and passing through the disk's center, which is then analogous to the North Pole. Gravity points down into the disk. (We consider the effects of sphericity in the next section; until then, Earth is flat.) Let Earth's angular velocity be X. (Earth rotates around its axis about once a day, so its angular velocity is 2r radians per day and thus X ~ 7.27 X 10-5 radians per second.) The velocity of the surface of Earth is Xr, where r is the distance from the axis of rotation, so that a missile moving along a line of latitude (i.e., around the disk) with velocity u relative to Earth has a total velocity, U, of Xr + u. The total centrifugal force per unit mass experienced by the missile is then given by
The first term on the right-hand side, X2r, is the centrifugal force due to the rotation of Earth itself. The second term is the additional centrifugal force due to the additional velocity of the missile. The third term, 2Xu, is the coriolis force; for oceanography and meteorology, it is the most important term of the three. Why so? The first term, X2r is a constant, and as we noted previously, its effects on Earth can be incorporated into a slightly changed gravitational term. The ratio of size of the second term to the third term is u:2Xr, and given that 2Xr ~ 900m s-1, the ratio is almost always much smaller than unity for winds and ocean currents. The factor 2X arises so frequently that we give it the special symbol f, so that the coriolis force, per unit mass, on a body is equal to f times its speed. The Coriolis force on a body initially moving meridionally (toward the axis of rotation) is given by the same expression (we show this expression explicitly in appendix A to this chapter), with the force acting to deflect the flow so that it begins to move in a zonal directionâ€”in other words, the Coriolis force acts at right angles to the direction the body is moving.
It is traditional in oceanography and meteorology to denote the zonal (eastward) velocity of the fluid by the symbol u, and the meridional (northward) velocity by v. Thus, the coriolis force is given by
Meridional Coriolis force = -fu. (3.4b)
Why is there a minus sign in the second equation? It is because a flow with a positive zonal velocity is deflected to the right in the Northern Hemisphere, toward the equator. The Coriolis force must therefore be negative to generate a negative meridional velocity.
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